Causal State Communication

The problem of state communication over a discrete memoryless channel with discrete memoryless state is studied when the state information is available strictly causally at the encoder. It is shown that block Markov encoding, in which the encoder communicates a description of the state sequence in the previous block by incorporating side information about the state sequence at the decoder, yields the minimum state estimation error. When the same channel is used to send additional independent information at the expense of a higher channel state estimation error, the optimal tradeoff between the rate of the independent information and the state estimation error is characterized via the capacity-distortion function. It is shown that any optimal tradeoff pair can be achieved via rate-splitting. These coding theorems are then extended optimally to the case of causal channel state information at the encoder using the Shannon strategy.